矩形的极小括号覆盖的构造
Construction of Minimal Bracketing Covers for Rectangles
摘要：
我们构造了二维单位立方体中矩形（锚定）集合系统的显式$δ$ -括号覆盖。更确切地说，这些$δ$包围覆盖的基数由$δ^{-2} + o(δ^{-2})$从上方界定。本文给出了$d$维锚定盒的任意$δ$ -包围覆盖的基数的一个下界. Gnewuch，Bracketing numbers for axis-parallel boxes and applications to geometric discrimination，J. Complexity 24（2008）154-172]暗示了维度$d=2$中的下界$δ^{-2}+O(δ^{-1})$，表明我们构造的覆盖是（基本上）最优的。我们还研究了矩形集合系统的其他$δ$括号覆盖，推导出其基数渐近展开中最重要项$δ^{-2}$的系数，并计算其显式值的基数$δ$。
英文摘要：
We construct explicit $δ$-bracketing covers with minimal cardinality for the set system of (anchored) rectangles in the two dimensional unit cube. More precisely, the cardinality of these $δ$-bracketing covers are bounded from above by $δ^{-2} + o(δ^{-2})$. A lower bound for the cardinality of arbitrary $δ$-bracketing covers for $d$-dimensional anchored boxes from [M. Gnewuch, Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy, J. Complexity 24 (2008) 154-172] implies the lower bound $δ^{-2}+O(δ^{-1})$ in dimension $d=2$, showing that our constructed covers are (essentially) optimal. We study also other $δ$-bracketing covers for the set system of rectangles, deduce the coefficient of the most significant term $δ^{-2}$ in the asymptotic expansion of their cardinality, and compute their cardinality for explicit values of $δ$.
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